Article,
A Note on a Characterization of the Exponential Distribution Based on a Type II Censored Sample
The Annals of Statistics, 23 (3): 769--773 (1995)
Abstract
Let X(1) ≤ X(2) ≤ ⋯ ≤ X(n) be the order statistics of a random sample of n lifetimes. The total-time-on-test statistic at X(i) is defined by Si,n = ∑i j = 1(n - j + 1)(X(j) - X(j - 1)), 1 ≤ i ≤ n. A type II censored sample is composed of the r smallest observations and the remaining n - r lifetimes which are known only to be at least as large as X(r). Dufour conjectured that if the vector of proportions (S1,n/Sr,n, ..., Sr - 1,n/Sr,n) has the distribution of the order statistics of r - 1 uniform(0, 1) random variables, then X1 has an exponential distribution. Leslie and van Eeden proved the conjecture provided n - r is no longer than (1/3)n - 1. It is shown in this note that the conjecture is true in general for n ≥ r ≥ 5. If the random variable under consideration has either NBU or NWU distribution, then it is true for n ≥ r ≥ 2, n ≥ 3. The lower bounds obtained here do not depend on the sample size.
